Targeting Individuals or Clusters With Unequal Sizes

Why the Target of Averaging Matters

When clusters have different numbers of members, “the average spillover effect” does not specify a unique quantity. An average over people gives a large cluster more influence because it contains more people. An average over clusters gives every cluster the same influence, whether it contains five people or five hundred. Both can be meaningful causal estimands, but they answer different questions.

This distinction matters when deciding how to summarize an intervention that changes other members’ treatment or exposure. A health program might ask how much the average person benefits when peers receive the program. A regional administrator might instead ask how much the average community benefits when a community-level policy changes. With equal-sized clusters these averages coincide; with unequal sizes they generally do not.

Set-Up and Definitions

Suppose a finite population is partitioned into \(J\) clusters. Cluster \(j\) contains \(n_j\) individuals, indexed by \(i=1,\ldots,n_j\), and the total number of individuals is \(N=\sum_{j=1}^{J}n_j\). Assume partial interference: an individual's outcome may depend on treatment or exposure within their own cluster, but not on assignments in other clusters. To define a spillover contrast, let \(Y_{ij}(a,g)\) denote the potential outcome for individual \(i\) in cluster \(j\) when their own treatment is \(a\) and the relevant within-cluster exposure is \(g\). For example, \(g\) may specify a peer-treatment saturation or an exposure category.

Compare two exposure conditions, \(g_1\) and \(g_0\), while holding own treatment at the same value \(a\). This isolates a spillover contrast: a difference associated with changing the exposure of others, not a contrast in the individual's own treatment. The exposure conditions must be well-defined for the population being studied. The formula below describes a finite-population causal estimand; identifying it from observed data additionally requires a suitable design or valid adjustment assumptions, including positivity for the exposure conditions being compared.

Definition: The individual spillover effect for person \(i\) in cluster \(j\) is \(\tau_{ij}(a;g_1,g_0)=Y_{ij}(a,g_1)-Y_{ij}(a,g_0)\). The cluster-specific mean spillover effect is \(\tau_j(a;g_1,g_0)=\frac{1}{n_j}\sum_{i=1}^{n_j}\tau_{ij}(a;g_1,g_0)\). The individual-population average spillover effect averages over all individuals, while the cluster-average spillover effect averages the cluster-specific means with equal weight across clusters.

Writing the two estimands explicitly makes their weighting transparent:

\tau_{\text{ind}}(a;g_1,g_0) = \frac{1}{N}\sum_{j=1}^{J}\sum_{i=1}^{n_j}\tau_{ij}(a;g_1,g_0) = \frac{\sum_{j=1}^{J}n_j\tau_j(a;g_1,g_0)}{\sum_{j=1}^{J}n_j}, \qquad \tau_{\text{cl}}(a;g_1,g_0) = \frac{1}{J}\sum_{j=1}^{J}\tau_j(a;g_1,g_0).

The individual-population estimand weights cluster \(j\)'s mean effect by \(n_j/N\). It is the effect averaged over a person selected uniformly from the population. The cluster-average estimand assigns weight \(1/J\) to each cluster. It is the effect averaged over a cluster selected uniformly from the set of clusters. These are different sampling perspectives and different target populations, not competing calculations of one universally correct answer.

How Cluster Size Changes the Average

The relationship between size and effect determines the difference between the two estimands. To see this, let \(\bar n=N/J\) be the mean cluster size, and define the finite-population covariance across clusters as \(\operatorname{Cov}_{J}(n_j,\tau_j)=\frac{1}{J}\sum_{j=1}^{J}(n_j-\bar n)(\tau_j-\tau_{\text{cl}})\). Then the individual-population effect can be written as the cluster-average effect plus a size-effect covariance term.

\tau_{\text{ind}} = \frac{\sum_j n_j\tau_j}{N} = \frac{\frac{1}{J}\sum_j n_j\tau_j}{\bar n} = \frac{\bar n\tau_{\text{cl}}+\operatorname{Cov}_{J}(n_j,\tau_j)}{\bar n} = \tau_{\text{cl}}+\frac{\operatorname{Cov}_{J}(n_j,\tau_j)}{\bar n}.

For the third equality, expand the covariance: \(\operatorname{Cov}_{J}(n_j,\tau_j)=\frac{1}{J}\sum_j n_j\tau_j-\bar n\tau_{\text{cl}}\). Substituting this expression gives the displayed result. Thus, larger clusters having larger effects makes the individual-weighted average exceed the cluster-weighted average; larger clusters having smaller effects makes it fall below. If all clusters have equal size, or if size and cluster effect have zero finite-population covariance, the two averages agree.

Worked Example: Two Averages From Three Clusters

Consider three hypothetical clusters with sizes 2, 4, and 10. Suppose the cluster-specific mean spillover effects, in outcome units per person, are 2, 0, and 5, respectively. These effects represent the mean within-cluster contrast between \(g_1\) and \(g_0\) at the same own-treatment value.

Individual-population average. There are \(N=2+4+10=16\) people. Weight each cluster effect by its number of people:

\tau_{\text{ind}} = \frac{2(2)+4(0)+10(5)}{16} = \frac{54}{16} = 3.375 \text{ outcome units per person}.

Cluster average. Give each of the three clusters equal weight:

\tau_{\text{cl}} = \frac{2+0+5}{3} = \frac{7}{3} \approx 2.333 \text{ outcome units per cluster mean}.

The individual-weighted effect is larger because the largest cluster also has the largest effect. The first result describes the average person's spillover contrast; the second describes the average of the three cluster-specific mean contrasts. They are not interchangeable, even though both use the same cluster effects.

From Potential Outcomes to a Policy Summary

The weighting choice should follow the policy question. If a policy changes peer exposure for people across a population and the outcome of interest is the average outcome per person, the natural summary is individual-weighted. If the policy is implemented at the cluster level and the decision-maker wants each community to count equally, the cluster-weighted summary may be more relevant. For a policy focused on total outcome change rather than a mean, neither average alone is sufficient: cluster totals or the total population effect may be the appropriate target.

A cluster-average effect is not automatically the effect of “targeting clusters,” and an individual-average effect is not automatically the effect of “targeting individuals.” The intervention and the averaging rule are separate choices. For example, a policy could assign the same exposure condition to every member of a selected cluster but evaluate outcomes averaged over all people; that is a cluster-level intervention with an individual-weighted outcome summary. State both what is changed and who receives equal weight in the final average.

Worked Example: Interpreting the Same Contrast for Two Audiences

Suppose a hypothetical evaluation has four clusters. Their sizes are 5, 5, 20, and 30 people, and their estimated cluster-specific mean spillover contrasts are 1, 3, 2, and 0 points on a health score. Treat these values as supplied cluster-level contrasts for illustration; the calculation alone does not establish that the contrasts are causally identified.

A public-health report asking for the average person's contrast uses the population size \(N=60\):

\widehat{\tau}_{\text{ind}} = \frac{5(1)+5(3)+20(2)+30(0)}{60} = \frac{60}{60} = 1.00 \text{ health-score point per person}.

A council comparing the average community-level mean contrast weights all four clusters equally:

\widehat{\tau}_{\text{cl}} = \frac{1+3+2+0}{4} = 1.50 \text{ health-score points per cluster mean}.

The two statements are both accurate but describe different averages. The larger clusters have lower contrasts than the smaller clusters in this example, so equal cluster weighting produces the larger number. A report should name the target explicitly, rather than presenting either estimate as simply “the average effect.”

Sampling and Estimation Implications

The weighting distinction also appears in how data are sampled and analyzed. If one person is sampled uniformly from the population, the probability that the person comes from cluster \(j\) is \(n_j/N\). Consequently, clusters are encountered in proportion to their size, and an average over sampled people corresponds to the individual-population target when the sampling and estimation procedures are appropriate. By contrast, selecting a cluster uniformly gives each cluster probability \(1/J\); averaging cluster means from such a sample targets the cluster-average quantity.

In an experiment or observational analysis, estimated cluster effects may have unequal precision as well as unequal sizes. Precision-based weighting can be useful for estimating a specified target, but it does not by itself define the causal estimand. The analyst must distinguish the weights used to represent the target population from weights used for estimation or variance reduction. For example, weighting cluster contrasts by their inverse variances generally targets a precision-weighted combination unless the procedure is constructed to recover the desired individual- or cluster-weighted estimand.

Likewise, unequal cluster sizes do not alone justify weighting. If cluster effects are identical, the two point estimands coincide regardless of the sizes. If effects differ, weighting is consequential. A valid causal interpretation still depends on whether the exposure contrast is identified: consistency of the exposure definition, appropriate control of confounding in an observational setting or the relevant randomization in an experiment, and positivity for each compared exposure condition. Choosing the correct average does not repair a failure of identification.

Worked Example: A Uniform Cluster Sample Versus a Uniform Person Sample

Imagine a hypothetical population with two clusters: a small cluster of 10 people with a cluster-specific spillover effect of 4 points, and a large cluster of 90 people with an effect of 1 point. A person selected uniformly from the population has a \(10/100\) chance of coming from the small cluster and a \(90/100\) chance of coming from the large cluster. Therefore the individual-population effect is

\tau_{\text{ind}} = \frac{10}{100}(4)+\frac{90}{100}(1) = 0.4+0.9 = 1.30 \text{ points per person}.

If instead a cluster is selected uniformly, each cluster has probability \(1/2\), so the average cluster effect is

\tau_{\text{cl}} = \frac{1}{2}(4)+\frac{1}{2}(1) = 2.50 \text{ points per cluster mean}.

The larger cluster has a smaller effect, which pulls the person-weighted average toward 1 point. The cluster-weighted average gives the small cluster the same influence as the large cluster. Thus, the sampling rule and the intended policy summary need to be aligned rather than treated as incidental details.

Common Mistakes

Calling one weighting rule universally correct. The correction is to state the target: an average over individuals or an average over clusters.
Assuming a cluster mean is a cluster total. Equal weighting of cluster means does not give equal weight to each person's outcome and does not summarize total population impact.
Using cluster size weights without defining the estimand. Size weights are appropriate for the individual-population average when the cluster effects are the relevant components; they are not a general-purpose fix for unequal cluster sizes.
Interpreting a difference between weighted averages as a different causal mechanism. The difference can arise solely from size-effect association, even when the underlying cluster-specific effects are unchanged.
Assuming a numerical estimate is causal because it has a causal label. The averaging formula determines the target, but causal identification still requires a valid design or assumptions.

Key takeaway: With unequal cluster sizes, the individual-population spillover effect is a size-weighted average of cluster effects, while the cluster-average spillover effect weights every cluster equally. Choose the estimand to match the policy question, and report the weighting rule and target population clearly.

Check Your Understanding

Use the definitions and weighting rules above to answer the following.

In your own words, what population is represented by the individual-population average spillover effect?
Two clusters have sizes 5 and 15 and cluster-specific effects of 4 and 1. Calculate the individual-population and cluster-average effects.
Under what conditions do the individual-population and cluster-average estimands agree?
If larger clusters tend to have smaller spillover effects, how does the individual-weighted average tend to compare with the cluster-weighted average?
Why does choosing size weights not, by itself, establish that an observational spillover contrast is causally identified?